 Molecular-weight scaling of an ideal-chain gyration radius

For a long ideal chain of $N$ independent statistical steps of length $b$, $R_g^2=N{b}^2/6$. With fixed chain chemistry and $b$, the molar mass $M$ is proportional to $N$. Thus multiplying molar mass by $s$ multiplies the radius of gyration by $\sqrt{s}$; solvent quality can change both the statistical segment length and the size exponent.